A Very Simple Introduction to Matlab
Aaron N. K. Yip
01-2015
MATLAB is an interactive, matrix-based system for scientific and engineering numerical computations and visualizations. MATLAB is the short form for MATrix LABoratory.
MATLAB is very user-friendly as a desk calculator, computational tool for linear algebra and
matrix analysis, solver for ordinary and partial differential equations, graph plotter and much more.
1
Get Onto Matlab
Matlab is becoming more and more popular. It can be found in many PC and UNIX systems.
• In the various campus instructional computer labs, you can access Matlab by simply clicking
its icon. (It can be found by using find.)
• Matlab also exists in most UNIX systems. Just type matlab to see if it exists in the system
you are using.
There should be a person on duty in the Computing Center if you have problems accessing
Matlab.
2
Try This....
The following is the actual screen you will see (plus some edited comments). For starters, after you
have opened matlab, you can just follow the lines and see.... Jump to the next section at anytime
if you wish.
>>
% This is the matlab prompt.
>> A=[1 2 3; 4 5 6; 7 8 9]
% Creating a matrix called A.
A =
1
4
7
2
5
8
3
6
9
1
>> B=[
1 2 3
5 6 1
3 -1 9
]
% Creating a matrix called B.
% Note that the format is quite flexible
B =
1
5
3
2
6
-1
3
1
9
>> A
% To see what is A.
A =
1
4
7
2
5
8
3
6
9
>> A + B
% A + B = ?
ans =
2
9
10
4
11
7
6
7
18
>> C = A + B
% Set C = A + B
C =
2
9
10
4
11
7
6
7
18
2
>> D= 2*A - 3*B
% Compute D = 2*A - 3*B
D =
-1
-7
5
-2
-8
19
-3
9
-9
>> C=[1; 2; 3]
% Create a column vector
C =
1
2
3
>> R = [ 1; 2; 3]
% Create a row vector
R =
1
2
3
>> R=[1 2 3]
R =
1
2
3
>> whos
Name
A
B
C
Size
% You can see what variables you have
% defined and their data.
Elements
Bytes
Density
Complex
3 by 3
3 by 3
3 by 1
9
9
3
3
72
72
24
Full
Full
Full
No
No
No
D
R
ans
3 by 3
1 by 3
3 by 3
9
3
9
72
24
72
Full
Full
Full
No
No
No
Grand total is 42 elements using 336 bytes
>> A
% To see A again.
A =
1
4
7
2
5
8
3
6
9
>> A(1,1)
% You can see particular entry of A
ans =
1
>> A(3,2)
% the (3,2) entry of A is?
ans =
8
>> A(2,2)=3.5
% change the (2,2) entry of A to be 3.5
A =
1.0000
4.0000
7.0000
>> A(1,3)=5.1
2.0000
3.5000
8.0000
3.0000
6.0000
9.0000
% change the (1,3) entry of A to be 5.1
A =
4
1.0000
4.0000
7.0000
2.0000
3.5000
8.0000
5.1000
6.0000
9.0000
>> P=rand(4)
% create a random 4 by 4 matrix
P =
0.2470
0.9826
0.7227
0.7534
0.6515
0.0727
0.6316
0.8847
0.2727
0.4364
0.7665
0.4777
>> Q=rand(3)
0.2378
0.2749
0.3593
0.1665
% create a random 3 by 3 matrix
Q =
0.4865
0.8977
0.9092
>> rank(A)
0.0606
0.9047
0.5045
0.5163
0.3190
0.9866
% To find the rank of A
ans =
3
>> help rand
% online help for rand
RAND Uniformly distributed random numbers and matrices.
RAND(N) is an N-by-N matrix with random entries, ordinarily
chosen from a uniform distribution on the interval (0.0,1.0).
RAND(M,N) or RAND([M,N]) is an M-by-N matrix with random entries.
RAND(SIZE(A)) is the same size as A.
RAND with no arguments is a scalar whose value changes each time
it is referenced.
5
RAND(’seed’) returns the current seed of the uniform generator.
RAND(’seed’,s) sets the uniform generator seed to s.
RAND(’seed’,0) resets the seed its startup value.
RAND(’seed’,sum(100*clock)) sets it to a different value each time.
By default, RAND samples a uniform distribution. The function
RANDN generates normally distributed random matrices.
RAND and RANDN have separate generators, each with its own seed.
Previous versions of MATLAB allowed RAND(’normal’) to switch
the prevailing distribution to normal, RAND(’uniform’) to switch
back to uniform distribution, and RAND(’dist’) to return a string
containing the prevailing distribution, either ’uniform’ or ’normal’.
MATLAB Version 4.0 continues to allow this switch, but
issues a warning message discouraging its use.
See also RANDN, SPRANDN.
>> help rank
% online help for rank
RANK Number of linearly independent rows or columns.
K = RANK(X) is the number of singular values of X
that are larger than MAX(SIZE(X)) * NORM(X) * EPS.
K = RANK(X,tol) is the number of singular values of X that
are larger than tol.
>> 1+2
% simple arithmetics
ans =
3
>> 2*7
ans =
6
14
>> 5/2
ans =
2.5000
>> pi
ans =
3.1416
>> exp(1)
ans =
2.7183
>> format long
>> exp(1)
% change the format to be long
ans =
2.71828182845905
>> help format
FORMAT Set output format.
All computations in MATLAB are done in double precision.
FORMAT may be used to switch between different output
display formats as follows:
FORMAT
Default. Same as SHORT.
FORMAT SHORT
Scaled fixed point format with 5 digits.
FORMAT LONG
Scaled fixed point format with 15 digits.
FORMAT SHORT E Floating point format with 5 digits.
7
FORMAT LONG E
FORMAT HEX
FORMAT +
Floating point format with 15 digits.
Hexadecimal format.
The symbols +, - and blank are printed
for positive, negative and zero elements.
Imaginary parts are ignored.
FORMAT BANK
Fixed format for dollars and cents.
FORMAT COMPACT Suppress extra line-feeds.
FORMAT LOOSE
Puts the extra line-feeds back in.
FORMAT RAT
Approximation by ratio of small integers.
>> Q
Q =
0.48651738301223
0.89765628655332
0.90920810164381
0.06056432754759
0.90465309233621
0.50452289474407
0.51629196364260
0.31903294116214
0.98664211201791
2.00000000000000
3.50000000000000
8.00000000000000
5.10000000000000
6.00000000000000
9.00000000000000
>> 3/4
ans =
0.75000000000000
>> A
A =
1.00000000000000
4.00000000000000
7.00000000000000
>> format
>> A
% reset format to be default - short
A =
8
1.0000
4.0000
7.0000
2.0000
3.5000
8.0000
5.1000
6.0000
9.0000
>> A*B
% A*B, matrix multiplication
ans =
26.3000
39.5000
74.0000
8.9000
23.0000
53.0000
50.9000
69.5000
110.0000
>> B*A
% B*A, matrix multiplication
% note that A*B is not equal to B*A
ans =
30.0000
36.0000
62.0000
33.0000
39.0000
74.5000
44.1000
70.5000
90.3000
>> C
C =
1
2
3
>> A*C
ans =
20.3000
29.0000
50.0000
>> C*A
% Matrix multiplication fails due to
9
% incompatible dimensions
??? Error using ==> *
Inner matrix dimensions must agree.
>> A^2
% A^A = A*A
ans =
44.7000
60.0000
102.0000
49.8000
68.2500
114.0000
63.0000
95.4000
164.7000
49.8000
68.2500
114.0000
63.0000
95.4000
164.7000
>> A*A
ans =
44.7000
60.0000
102.0000
>> A^3
ans =
1.0e+03 *
0.6849
1.0008
1.7109
0.7677
1.1221
1.9206
1.0938
1.5741
2.6865
0.7677
1.0938
>> A*A*A
ans =
1.0e+03 *
0.6849
10
1.0008
1.7109
1.1221
1.9206
1.5741
2.6865
>> B
B =
1
5
3
2
6
-1
3
1
9
>> B’
% transpose
ans =
1
2
3
5
6
1
3
-1
9
>> B’’
% B transpose and transpose again
ans =
1
5
3
2
6
-1
3
1
9
>> C
C =
1
2
3
>> R
11
R =
1
2
3
>> C*R
% column vector times row vector
ans =
1
2
3
2
4
6
3
6
9
>> R*C
% row vector times column vector
ans =
14
>> zeros(3)
% To create a 3 by 3 zero matrix
ans =
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
>> zeros(5)
ans =
0
0
0
0
0
>> eye(5)
0
0
0
0
0
0
0
0
0
0
% To create a 5 by 5 identity matrix
12
ans =
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
1
>> eye(3)
ans =
1
0
0
>> E=rand(3,4)
% To create a random 3 by 4 matrix
E =
0.4940
0.2661
0.0907
0.9478
0.0737
0.5007
0.3841
0.2771
0.9138
0.5297
0.4644
0.9410
>> rref(E)
% To obtain the reduced row echelon form
% of E
ans =
1.0000
0
0
0
1.0000
0
0
0
1.0000
0.6739
-0.2346
1.0914
>> B
B =
13
1
5
3
2
6
-1
3
1
9
0
1
0
0
0
1
>> rref(B)
ans =
1
0
0
>> help rref
% Online help for rref
RREF Reduced row echelon form.
R = RREF(A) produces the reduced row echelon form of A.
[R,jb] = RREF(A) also returns a vector, jb, so that:
r = length(jb) is this algorithm’s idea of the rank of A,
x(jb) are the bound variables in a linear system, Ax = b,
A(:,jb) is a basis for the range of A,
R(1:r,jb) is the r-by-r identity matrix.
[R,jb] = RREF(A,TOL) uses the given tolerance in the rank tests.
Roundoff errors may cause this algorithm to compute a different
value for the rank than RANK, ORTH and NULL.
See also RREFMOVIE, RANK, ORTH, NULL, QR, SVD.
>> lookfor row
% lookfor keyword row
RANK Number of linearly independent rows or columns.
RREF Reduced row echelon form.
RREFMOVIE Movie of the computation of the reduced row echelon form.
WATCHOFF Sets the current figure pointer to the arrow.
FINDROW Find the rows of a matrix that match the input string.
IDUIPOIN Sets and resets the window pointer to watch and arrow
14
ARROW an arrow
PDEBSPLIT Splits space separated string into separate rows.
>> lookfor echelon
RREF Reduced row echelon form.
RREFMOVIE Movie of the computation of the reduced row echelon form.
>> B
B =
1
5
3
2
6
-1
3
1
9
>> det(B)
% determinant of B
ans =
-98
>> inv(B)
% inverse of B
ans =
-0.5612
0.4286
0.2347
0.2143
0
-0.0714
0.1633
-0.1429
0.0408
>> D=inv(B)
% Set D = inv(B)
D =
-0.5612
0.4286
0.2347
0.2143
0
-0.0714
0.1633
-0.1429
0.0408
15
>> B*D
% B*D = I, of course
ans =
1.0000
-0.0000
0
0.0000
1.0000
0.0000
0
0.0000
1.0000
>> D*B
% D*B = I, of course
ans =
1.0000
0.0000
-0.0000
-0.0000
1.0000
0.0000
-0.0000
0.0000
1.0000
>> det(D)
% det(D) = ?
ans =
-0.0102
>> det(B)*det(D)
% det(B)*det(D) = 1, of course
ans =
1.0000
>> whos
Name
A
B
C
D
E
P
Size
3
3
3
3
3
4
by
by
by
by
by
by
Elements
Bytes
Density
9
9
3
9
12
16
72
72
24
72
96
128
Full
Full
Full
Full
Full
Full
3
3
1
3
4
4
16
Complex
No
No
No
No
No
No
Q
R
ans
3 by 3
1 by 3
1 by 1
9
3
1
72
24
8
Full
Full
Full
No
No
No
Grand total is 71 elements using 568 bytes
>> size(P)
% size of P
ans =
4
4
>> size(R)
% size of R
ans =
1
3
>> help size
SIZE Matrix dimensions.
D = SIZE(X), for M-by-N matrix X, returns the two-element
row vector D = [M, N] containing the number of rows and columns
in the matrix.
[M,N] = SIZE(X) returns the number of rows and columns
in separate output variables.
M = SIZE(X,1) returns just the number of rows.
N = SIZE(X,2) returns just the number of columns.
>> save myfile.mat
% save the workspace into file called
% myfile.mat
>> whos
Name
A
Size
Elements
Bytes
Density
9
72
Full
3 by 3
17
Complex
No
B
C
D
E
P
Q
R
ans
3
3
3
3
4
3
1
1
by
by
by
by
by
by
by
by
3
1
3
4
4
3
3
2
9
3
9
12
16
9
3
2
72
24
72
96
128
72
24
16
Full
Full
Full
Full
Full
Full
Full
Full
No
No
No
No
No
No
No
No
Grand total is 72 elements using 576 bytes
>> clear
>> whos
% clear the workspace.
% now, there are no variables defined.
>> load myfile
% don’t worry! You can reload the
% previous workspace
>> whos
Name
A
B
C
D
E
P
Q
R
ans
% See, now all the previous variables
% are there.
Elements
Bytes
Density
Complex
Size
3
3
3
3
3
4
3
1
1
by
by
by
by
by
by
by
by
by
3
3
1
3
4
4
3
3
2
9
9
3
9
12
16
9
3
2
Grand total is 72 elements using 576 bytes
>> quit
18
72
72
24
72
96
128
72
24
16
Full
Full
Full
Full
Full
Full
Full
Full
Full
No
No
No
No
No
No
No
No
No
3
Some Useful Commands
It is quite useful to get acquainted with the following. Just try them out.... You can get more
information about them by typing help command.
3.1
General Commands
lookfor keyword
look for the commands related to a key word. This is very useful if you forget the exact name
of the command.
help command —
give online help for the command (if you know the name)
diary filename —
save all the screen output to the file “filename”. Default filename is “diary”. diary off will
turn off the diary mode. After you are done, you can then take a look of the screen output
(and edit it) at your leisure.
save filename —
save the whole workspace (i.e. all the variables) into a file. Default filename is “matlab.mat”.
load filename —
load all the variables from the file “filename” into the workspace. Default filename is “matlab.mat”.
whos
list information about current variables. The long form of who.
clear
clear the workspace. You can clear a particular variable by typing clear variablename.
3.2
Mathematical Operations and Functions
You can just type help to get a list of all the types of functions available. Of particular interests
to starters are:
Operators and special characters.
+
- Plus
19
*
/
’
=
%
-
Minus
Matrix multiplication
division
Matrix Transpose
Assignment
Comment
Matrix analysis.
rref
det
inv
- Reduced row echelon form.
- Determinant.
- Matrix inverse.
rank
trace
null
orth
expm
-
Number of linearly independent rows or columns.
Sum of diagonal elements.
Null space.
Orthogonalization.
Matrix exponential.
Eigenvalues.
eig
poly
- Eigenvalues and eigenvectors.
- Characteristic polynomial.
Trigonometric.
sin
sinh
asin
asinh
cos
cosh
acos
acosh
tan
-
Sine.
Hyperbolic sine.
Inverse sine.
Inverse hyperbolic sine.
Cosine.
Hyperbolic cosine.
Inverse cosine.
Inverse hyperbolic cosine.
Tangent.
20
tanh
atan
atan2
atanh
sec
sech
asec
asech
csc
csch
acsc
acsch
cot
coth
acot
acoth
-
Hyperbolic tangent.
Inverse tangent.
Four quadrant inverse tangent.
Inverse hyperbolic tangent.
Secant.
Hyperbolic secant.
Inverse secant.
Inverse hyperbolic secant.
Cosecant.
Hyperbolic cosecant.
Inverse cosecant.
Inverse hyperbolic cosecant.
Cotangent.
Hyperbolic cotangent.
Inverse cotangent.
Inverse hyperbolic cotangent.
-
Exponential.
Natural logarithm.
Common logarithm.
Square root.
-
Absolute value.
Phase angle.
Complex conjugate.
Complex imaginary part.
Complex real part.
Exponential.
exp
log
log10
sqrt
Complex.
abs
angle
conj
imag
real
Numeric.
fix
floor
ceil
- Round towards zero.
- Round towards minus infinity.
- Round towards plus infinity.
21
round
rem
sign
- Round towards nearest integer.
- Remainder after division.
- Signum function.
Polynomials.
roots
poly
polyval
polyvalm
residue
polyder
conv
deconv
-
Find polynomial roots.
Construct polynomial with specified roots.
Evaluate polynomial.
Evaluate polynomial with matrix argument.
Partial-fraction expansion (residues).
Differentiate polynomial.
Multiply polynomials.
Divide polynomials.
22